Direct Vent fireplaces do not require a chimney. They are able to vent horizontally out a sidewall or vertically to the roof. With completely enclosed chambers that draw in air for combustion from the outside and expelling gasses to the outside, they allow you to install a fireplace in rooms that would not be practical for a more traditional fireplace.
Vent Free fireplaces lack any external vent. They are extremely simple to install and are entirely self-contained, not requiring any vents of chimneys. They tend to be around 99% energy efficient, and they produce no harmful by-products, converting the vast majority of their fuel to heat.
B-Vent fireplaces are some of the simplest designs on the market. They rely on the buoyancy of hot air to passively exhaust itself up and out of the ventilation pipes. They are often the least expensive units on the market, but they are usually less fuel efficient.